# Dynamics of Cycles in Polyhedra I: The Isolation Lemma

@article{Kessler2020DynamicsOC, title={Dynamics of Cycles in Polyhedra I: The Isolation Lemma}, author={Jan Kessler and Jens M. Schmidt}, journal={ArXiv}, year={2020}, volume={abs/2002.07698} }

A cycle $C$ of a graph $G$ is \emph{isolating} if every component of $G-V(C)$ is a single vertex. We show that isolating cycles in polyhedral graphs can be extended to larger ones: every isolating cycle $C$ of length $6 \leq |E(C)| < \left \lfloor \frac{2}{3}(|V(G)|+4) \right \rfloor$ implies an isolating cycle $C'$ of larger length that contains $V(C)$. By "hopping" iteratively to such larger cycles, we obtain a powerful and very general inductive motor for proving long cycles and computing…

## One Citation

On Tutte cycles containing three prescribed edges

- Mathematics
- 2021

A cycle C in a graph G is called a Tutte cycle if, after deleting C from G, each component has at most three neighbors on C. Tutte cycles play an important role in the study of Hamiltonicity of…

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